from this information, i can easily calculate that there is around a 7% chance of 5 teams advancing to the tournament, and less than a 1% chance of only 1 team advancing. i can also, of course, calculate that the expected number of teams advancing is 3.19.

what i'm having trouble with, however, is how to figure out how often exactly 2 teams, exactly 3 teams, and exactly 4 teams will advance. aside from manually multiplying out all of the possibilities and adding them together, which i'd rather avoid, is there an equation that accomplishes this?